The population, P, of a town t years after 1 January 2025 is modelled by P = 2400e0.05t
Write down the population of the town on 1 January 2025 according to the model.
Find the population predicted by the model after 10 years.
3 marks
Show answer
a) 2400b) 3957 (about 3960)
2
The value, £V, of a car t years after it was bought is modelled by V = 18 000e−0.15t
Find how long it takes for the value of the car to halve, giving your answer in years to 3 significant figures.
3 marks
Show answer
4.62 years
3
The curve C has equation y = 3e2x
Find dydx
Find the exact gradient of C at the point where x = ln 2
Find an equation of the tangent to C at the point where x = 0
4 marks
Show answer
a)dydx = 6e2xb) 24c) y = 6x + 3
4
The mass, m grams, of a radioactive sample t days after it was made is modelled by m = 40e−0.02t
Show that dmdt = −0.02m
Find the rate at which the mass is decreasing when t = 50
Find the value of t when the mass is decreasing at 0.5 grams per day.
7 marks
Show answer
a)dmdt = -0.8e-0.02t = -0.02mb) 0.294 g per dayc) t = 23.5 days
5
The number of bacteria, N, in a sample t hours after it is prepared is modelled by N = Aekt, where A and k are constants.
When t = 0 there are 500 bacteria and when t = 4 there are 800 bacteria.
Find the value of A and show that k = 14 ln 1.6
Use the model to predict the number of bacteria when t = 10
Give a reason why the model may not be reliable for large values of t
6 marks
Show answer
a) A = 500, k = 14ln 1.6b) 1619c) the model grows without limit, but food and space will run out
6
Two variables x and y satisfy y = axn, where a and n are constants.
Show that log10y = log10a + nlog10x
A graph of log10y against log10x is a straight line through the points (0, 0.6) and (2, 2.1)
Find the value of n and the value of a, giving a to 3 significant figures.
5 marks
Show answer
a) log y = log (axn) = log a + log xn = log a + n log xb) n = 0.75, a = 3.98
7
The value, £V, of a machine t years after it was bought is modelled by V = abt, where a and b are constants.
A graph of log10V against t is a straight line that crosses the vertical axis at 4.2 and has gradient −0.08
Write down an equation for the line.
Find the value of a and the value of b, giving each to 3 significant figures.
Interpret the value of b in the context of the model.
6 marks
Show answer
a) log V = 4.2 - 0.08tb) a = 15 800, b = 0.832c) each year the value is multiplied by 0.832, so it falls by about 16.8% a year
8
A cup of tea is left to cool. Its temperature, θ °C, t minutes after it was made is modelled by
θ = 20 + 75e−0.04t
Find the temperature of the tea when it was made.
Find the temperature of the tea after 10 minutes.
Find the time it takes for the tea to cool to 50 °C
State the temperature that the model suggests the tea will approach in the long term, and what it represents.
7 marks
Show answer
a) 95 °Cb) 70.3 °Cc) 22.9 minutesd) 20 °C, the temperature of the room
9
A student measures the time, T seconds, for one swing of a pendulum of length L metres. She plots log10T against log10L and finds that the points lie close to a straight line through (−1, −0.198) and (0.6, 0.602)
Explain why this suggests a model of the form T = aLn, where a and n are constants.
Find the value of n and the value of a, giving a to 3 significant figures.
Use the model to find the time for one swing of a pendulum of length 2.5 m
6 marks
Show answer
a) log T = log a + n log L, a straight line in log T and log Lb) n = 0.5, a = 2.00c) 3.17 s
10
The number of members, P, of an online chess club t months after it started is shown in the table.
t
1
2
3
5
P
162
219
295
538
log10P
2.2095
2.4698
2.7308
The club secretary suggests the model P = abt, where a and b are constants.
Show that this model gives log10P = log10a + tlog10b
Complete the table, giving the value to 4 decimal places.
Using the values for t = 1 and t = 5, find the value of a and the value of b, giving each to 3 significant figures.
Use the model to predict the number of members after 8 months, and comment on the reliability of this prediction.
9 marks
Show answer
a) log P = log a + log bt = log a + t log bb) 2.3404c) a = 120, b = 1.35d) about 1320; t = 8 is outside the data, so this is extrapolation and may be unreliable
11
The area, A m2, of a pond covered by algae t days after it is first measured is modelled by
A = 50ekt
where k is a constant. When t = 0, the area is increasing at a rate of 4 m2 per day.
Show that k = 0.08
Find the area covered when it is increasing at a rate of 20 m2 per day.
Find the value of t when this happens.
The pond has a surface area of 600 m2. Find the time when the model predicts the pond will be completely covered, and explain why the model cannot be used after that time.
8 marks
Show answer
a)dAdt = 50kekt, at t = 0: 50k = 4, k = 0.08b) 250 m2c) t = 20.1d) t = 31.1 days; after that the model predicts more algae than the pond can hold
12
In a forest, the number of grey squirrels, G, and the number of red squirrels, R, t years after the start of a study are modelled by
G = 300e0.04tR = 900e−0.02t
Show that the two populations are equal when t = 503 ln 3
Find the size of each population at this time.
Show that, at this time, the grey squirrel population is increasing at exactly twice the rate at which the red squirrel population is decreasing.
State one criticism of the model for the red squirrels in the long term.
9 marks
Show answer
a) 300e0.04t = 900e-0.02t, e0.06t = 3, t = 503ln 3b) 624c)dGdt = 0.04G, dRdt = -0.02R and G = R, so the rates are in the ratio 2 : 1d) e.g. the model never reaches 0, so it cannot show the red squirrels dying out; or it ignores conservation work
Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.